A SUMMARY OF THE MATHEMATICAL WORK AND PUBLICATIONS OF K. O. FRIEDRICHS AUTOBIOGRAPHICAL SKETCH OF K. O. FRIEDRICHS

Note: K. O. Friedrichs wrote this document in December 1978 as his own summary review of his primary work. It was followed by a list of his 178 publications written between 1927-1981 (including 95 journal articles, 38 lecture series, 37 reports and 8 separate books “ Click here to go to this complete list, including Major Awards and Honors.) This OCR (optical character read) edition was created by Martin Friedrichs in 2001 and required some interpretation of where to inset dates

The main area of my research was the theory of partial differential equations; in particular of equations that represent laws of physics or engineering science (1927)

My first two publications concern an invariant formulation of Newton’s law of gravitation and the existence theory for the equations of elastic plates. (Ph.D. Thesis).

Among other work done in Göttingen in the late nineteen hundred twenties, I mention the work on the initial problem for linear hyperbolic equations (together with Hans Lewy) (1927) and the work done with H. Lewy and R. Courant on partial difference equations. (1928)

One of the points brought out in the latter paper was that one cannot replace a hyperbolic-differential equation by a difference equation in an arbitrary manner and expect that the solutions of the latter equations approach those of the first one. The ratio of the time difference to the space difference must be sufficiently small. This observation (made by H. Lewy) played a considerable role in later years when hyperbolic-differential equations were computed approximately with the aid of difference equations.

Among the subjects of applied mathematics in subsequent years a remark about the possibility of transforming the minimum of an energy integral into the maximum of another such integral may be mentioned (1929). This remark is still on occasion referred to.

My most significant work in applied mathematics, done with J. J. Stoker in the United States, was the work on the nonlinear boundary problem of the buckled plate. The problem concerned a circular plate subject to uniform compressive forces acting on the edge. If these forces are increased beyond a certain value the plate will buckle, as is well known. Our problem was to investigate what happens to the plate when the compressive forces are increased indefinitely. This was done by hand-computing and took many months. One feature found, which was surprising to us, was that eventually the radial stress at and near the center of the plate becomes a tensile one. This computational result (1939) was verified by an asymptotic-analysis. (In a concrete case this result has been verified experimentally some years ago.) (1941)

Somewhat later I published additional papers connected with plate theory. Two of them (1937)(1947), of a purely mathematical character, concern Korn’s inequality (one for two dimensions, one for any dimension). This inequality is needed in the theory of elastic plates with free boundaries.

In another paper the peculiar boundary conditions one may impose at the edge of a plate with a free boundary were derived by an asymptotic analysis, (with R. F. Dressler). (1949) (1950) (1961)

In the years after 1943 I did various investigations on Fluid Dynamics, partly together with R. Courant. This work led to the publication of the book on “Supersonic Flow and Shockwaves,” which appeared in 1948 and was widely used by mathematically minded aero-dynamicists.

In a report “On the Mathematical Theory of De-flagration and Detonations” arguments are used which are related to the “boundary layer analysis” introduced by me somewhat earlier. This paper was reprinted in the “Lectures on Combustion Theory”, Courant Institute 1978.

There are various additional papers that belong more or less to applied mathematics, other than fluid dynamics.

In one of these papers (1937) the theory of perturbation of continuous spectra was initiated and later applied to problems of quantum theory. (1948)

In a paper on nonlinear oscillations (with W. Wasow), concerned with asymptotic considerations, the notion of “singular perturbation” was introduced. (1946)

Various asymptotic phenomena in mathematical physics were presented in the Josiah Willard Gibbs lecture given on invitation by the American Mathematical Society. (1955)

Among earlier purely mathematical work was a series of papers (1934)(1935)on the spectral theory of semi bounded linear operators in which von Neumann’s Hilbert space theory is effectively used. In a later paper (1939) on differential operators of the first order the notions of weak and strong solutions are introduced. It is proved, for the case of constant coefficients, that the weak and the strong solutions are identical. For this proof a particular class of transformations, later called mollifiers, is employed.

In a subsequent paper (1944) it is shown that this tool is strong enough to prove the identity of the weak and the strong solutions, also for first order operators with non-constant coefficients.

In all these considerations neither Lebesgue theory nor distribution theory is used.

Perhaps the most significant work in the years after 1937 was on a class of equation that I introduced and called symmetric hyperbolic linear differential equations. (1954) These are systems of the first order; the symmetry involved is that of the matrix of the coefficients of the differentiated terms. I proved the existence of the solution of the initial value problem. Again the treatment was based on general operator theory and again used modifiers. The equations have two striking properties: the eigenvectors associated with the matrix need not be distinct and the initial data need not have infinitely many derivatives but only a finite number. It is this class of equations to which essentially all non-degenerate equations of motion can be reduced.(1947)

A paper which is not immediately related to applied mathematics should be mentioned. It concerns the existence of differential forms on Riemannian manifolds. It is also based on the theory of linear operators. One feature of this work is the use of a basic inequality which is not covariant although the resultant differential forms are covariant. (l955)

The existence theory for “accretive” equations, which may be partly hyperbolic, partly elliptic, and partly of intermediate types, may be regarded as an extension of the previous work on elliptic and hyperbolic equations. An essential feature of this work was that one need not pay any attention to the places at which the type of the equation changes. (1958)

A boundary value problem for pseudo differential operators was treated, with P. D. Lax, in a paper of 1965. Additional material on such operators was presented in Courant Institute lecture notes of 1970.

In a series of five papers on quantum theory of fields an attempt was made to put this theory on a sound mathematical basis. (1951-53)

This attempt had only limited success. Still a few observations that resulted may be of some value. Among them is the observation that there are dif-ferent, non-equivalent realizations of the basic field operators. These five papers reappeared in book on Mathematical Aspects of the Quantum Theory of Fields. (1953) A somewhat different approach to this theory is presented in a book on Perturbation of Spectra in Hilbert Space (1967-second printing)

There are a number of papers on fluid dynamics of which only few will be mentioned.

One paper, with D. H. Hyers, is on the existence of the solitary wave. (1954)

Another work which appeared in lecture notes is the one on wave motion in magneto fluid dynamics. In particular it is shown that these equations can be written as symmetric-hyperbolic ones. The equations of relativistic magneto fluid dynamics were also formulated; but that they can be put into symmetric--hyperbolic-form was shown only in later work. (1974)

In a note written with P. D. Lax (1971) systems of conservation equations are treated which possess a convex extension. One of the statements in this paper is that such equations can be reduced to symmetric-hyperbolic ones.

A somewhat different treatment of such conservation equations is given by me in 1974. This treatment is applied in particular to the laws of relativistic electro magneto fluid dynamics. (1977) One of the results is the reduction of the equations of relativistic magneto fluid dynamics to symmetric-hyperbolic ones. Another result is the resolution of the controversy about the proper formulation of the energy momentum tensor of relativistic-electro magnetism.

In a paper “On the Notion of State in Quantum Mechanics” it is shown that the future values of unobserved observables can be determined in terms (1979) of their unobserved initial values.